Timeline for Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]
Current License: CC BY-SA 2.5
32 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2019 at 23:36 | comment | added | Timothy | stronger theory. Maybe we can deduce the consistency of ZFC from the consistency of ZF but we cannot prove the statement that ZFC is a true model of set theory. | |
| Jun 20, 2019 at 23:35 | comment | added | Timothy | Some people don't automatically accept the axiom of choice. They accept ZF but not ZFC. According to their claim, some other people have a strong intuition for the axiom of choice and will never understand why the axiom of choice might not actually be true. With their stubborn insistence on the axiom of choice, they claim that ZFC is consistent without a justifiable reason. They don't even try to see if they can deduce it from the consistency just of ZF. Indeed, we cannot prove the axiom of choice because it derives from ZFC and we cannot prove that ZFC is consistent because it derives from a | |
| Jun 27, 2010 at 4:36 | comment | added | Yemon Choi | Voting to close. While some of the answers have mentioned interesting things, I think the question itself starts from false premises and would really be better discussed on a blog comment thread (any suggestions?). But I am certainly willing to cast a vote for re-opening if people disagree and explain why. | |
| Jun 27, 2010 at 4:32 | history | closed |
Robin Chapman Andrew Stacey Harry Gindi darij grinberg Yemon Choi |
not constructive | |
| Jun 26, 2010 at 22:28 | answer | added | user7105 | timeline score: -2 | |
| Jun 23, 2010 at 14:37 | comment | added | Dan Piponi | I think this comment by Andrej Bauer is quite relevant. ZFC is merely an approximation to what mathematicians actually do. mathoverflow.net/questions/23060/set-theory-and-model-theory/… As a result, a problem in ZFC wouldn't bring mathematics to a crashing halt. | |
| Jun 23, 2010 at 13:51 | history | edited | Charles Matthews | CC BY-SA 2.5 |
question mark
|
| Jun 23, 2010 at 9:56 | history | edited | AgCl | CC BY-SA 2.5 |
deleted 55 characters in body
|
| Jun 22, 2010 at 19:08 | vote | accept | AgCl | ||
| Jun 22, 2010 at 19:04 | comment | added | Tom Goodwillie | @Pete: That's a widely held view, and I'm not really denying it. Let me revise my statement: If you're looking for utter certainty, then even mathematics is not entirely the right field. | |
| Jun 22, 2010 at 18:52 | answer | added | Kiochi | timeline score: 11 | |
| Jun 22, 2010 at 18:44 | answer | added | Emerton | timeline score: 17 | |
| Jun 22, 2010 at 18:38 | answer | added | Terry Tao | timeline score: 68 | |
| Jun 22, 2010 at 18:27 | comment | added | Paul Siegel | I can only speak for myself, but I don't worry about such things for the same reason that I still walk to work every day even though I could get hit by a car at any minute. If I spend the rest of my life trying to convince myself that what I think is a proof really is a proof and one day I actually succeed, then I will just wish I had spent all that time thinking about geometry instead. | |
| Jun 22, 2010 at 18:19 | answer | added | Igor Belegradek | timeline score: 6 | |
| Jun 22, 2010 at 18:18 | history | edited | AgCl | CC BY-SA 2.5 |
added 9 characters in body
|
| Jun 22, 2010 at 18:12 | answer | added | Pace Nielsen | timeline score: 10 | |
| Jun 22, 2010 at 18:02 | comment | added | Pete L. Clark | @Tom: my impression was that if one is looking for certainty, mathematics is the least wrong field. | |
| Jun 22, 2010 at 17:48 | answer | added | Jamie Banks | timeline score: 8 | |
| Jun 22, 2010 at 17:46 | comment | added | Tom Goodwillie | I dunno, if you're looking for certainty, maybe mathematics is the wrong field. | |
| Jun 22, 2010 at 17:31 | answer | added | mathy | timeline score: 6 | |
| Jun 22, 2010 at 16:34 | comment | added | François G. Dorais | I couldn't resist linking this comic - abstrusegoose.com/244 | |
| Jun 22, 2010 at 16:33 | history | edited | AgCl | CC BY-SA 2.5 |
added 64 characters in body; added 6 characters in body
|
| Jun 22, 2010 at 16:29 | comment | added | Brendan Cordy | Something to note: Supposing we could actually prove some consistency statement Con(ZFC) within ZFC, that's still no reason to believe that ZFC is consistent! Why? Well, suppose ZFC is inconsistent, then it would prove Con(ZFC) for sure! | |
| Jun 22, 2010 at 16:22 | history | edited | AgCl |
edited tags
|
|
| Jun 22, 2010 at 16:12 | comment | added | Carl Mummert | @Robin Chapman: in logic, at least, we routinely study the incompleteness theorems in great depth, but few if any logicians are worried about it or worried about its consequences. | |
| Jun 22, 2010 at 16:10 | answer | added | Carl Mummert | timeline score: 22 | |
| Jun 22, 2010 at 15:54 | history | edited | AgCl | CC BY-SA 2.5 |
deleted 7 characters in body
|
| Jun 22, 2010 at 15:49 | comment | added | Charles Matthews | In the version that there are recursively enumerable sets that are not recursive, that seems fair. Dieudonné certainly once said that if certain problems are not algorithmically soluble, then we should care more about other things. (But I disagree with the tenor of the question. If 0 = 1 results from some high-powered proof, that shifts the foundational debate back to a century ago. But some illumination will come out of it, as axiomatic set theory came out of the paradoxes.) | |
| Jun 22, 2010 at 15:42 | comment | added | Robin Chapman | Do mathematicians really "worry less and less each day about Godel's theorem"? | |
| Jun 22, 2010 at 15:37 | comment | added | AgCl | This is a rewording of a question which is deleted by peer pressure. I apologize for deleted comments, I could not recover them. I only recover the link for one of the suggested references: web.archive.org/web/20070205203647/http://www.hf.uio.no/ifikk/… | |
| Jun 22, 2010 at 15:36 | history | asked | AgCl | CC BY-SA 2.5 |